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Mirrors > Home > MPE Home > Th. List > Mathboxes > m1expevenALTV | Structured version Visualization version GIF version |
Description: Exponentiation of -1 by an even power. (Contributed by Glauco Siliprandi, 29-Jun-2017.) (Revised by AV, 6-Jul-2020.) |
Ref | Expression |
---|---|
m1expevenALTV | ⊢ (𝑁 ∈ Even → (-1↑𝑁) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqeq1 2739 | . . . 4 ⊢ (𝑛 = 𝑁 → (𝑛 = (2 · 𝑖) ↔ 𝑁 = (2 · 𝑖))) | |
2 | 1 | rexbidv 3177 | . . 3 ⊢ (𝑛 = 𝑁 → (∃𝑖 ∈ ℤ 𝑛 = (2 · 𝑖) ↔ ∃𝑖 ∈ ℤ 𝑁 = (2 · 𝑖))) |
3 | dfeven4 47563 | . . 3 ⊢ Even = {𝑛 ∈ ℤ ∣ ∃𝑖 ∈ ℤ 𝑛 = (2 · 𝑖)} | |
4 | 2, 3 | elrab2 3698 | . 2 ⊢ (𝑁 ∈ Even ↔ (𝑁 ∈ ℤ ∧ ∃𝑖 ∈ ℤ 𝑁 = (2 · 𝑖))) |
5 | oveq2 7439 | . . . . 5 ⊢ (𝑁 = (2 · 𝑖) → (-1↑𝑁) = (-1↑(2 · 𝑖))) | |
6 | neg1cn 12378 | . . . . . . . . 9 ⊢ -1 ∈ ℂ | |
7 | 6 | a1i 11 | . . . . . . . 8 ⊢ (𝑖 ∈ ℤ → -1 ∈ ℂ) |
8 | neg1ne0 12380 | . . . . . . . . 9 ⊢ -1 ≠ 0 | |
9 | 8 | a1i 11 | . . . . . . . 8 ⊢ (𝑖 ∈ ℤ → -1 ≠ 0) |
10 | 2z 12647 | . . . . . . . . 9 ⊢ 2 ∈ ℤ | |
11 | 10 | a1i 11 | . . . . . . . 8 ⊢ (𝑖 ∈ ℤ → 2 ∈ ℤ) |
12 | id 22 | . . . . . . . 8 ⊢ (𝑖 ∈ ℤ → 𝑖 ∈ ℤ) | |
13 | expmulz 14146 | . . . . . . . 8 ⊢ (((-1 ∈ ℂ ∧ -1 ≠ 0) ∧ (2 ∈ ℤ ∧ 𝑖 ∈ ℤ)) → (-1↑(2 · 𝑖)) = ((-1↑2)↑𝑖)) | |
14 | 7, 9, 11, 12, 13 | syl22anc 839 | . . . . . . 7 ⊢ (𝑖 ∈ ℤ → (-1↑(2 · 𝑖)) = ((-1↑2)↑𝑖)) |
15 | neg1sqe1 14232 | . . . . . . . . 9 ⊢ (-1↑2) = 1 | |
16 | 15 | oveq1i 7441 | . . . . . . . 8 ⊢ ((-1↑2)↑𝑖) = (1↑𝑖) |
17 | 1exp 14129 | . . . . . . . 8 ⊢ (𝑖 ∈ ℤ → (1↑𝑖) = 1) | |
18 | 16, 17 | eqtrid 2787 | . . . . . . 7 ⊢ (𝑖 ∈ ℤ → ((-1↑2)↑𝑖) = 1) |
19 | 14, 18 | eqtrd 2775 | . . . . . 6 ⊢ (𝑖 ∈ ℤ → (-1↑(2 · 𝑖)) = 1) |
20 | 19 | adantl 481 | . . . . 5 ⊢ ((𝑁 ∈ ℤ ∧ 𝑖 ∈ ℤ) → (-1↑(2 · 𝑖)) = 1) |
21 | 5, 20 | sylan9eqr 2797 | . . . 4 ⊢ (((𝑁 ∈ ℤ ∧ 𝑖 ∈ ℤ) ∧ 𝑁 = (2 · 𝑖)) → (-1↑𝑁) = 1) |
22 | 21 | rexlimdva2 3155 | . . 3 ⊢ (𝑁 ∈ ℤ → (∃𝑖 ∈ ℤ 𝑁 = (2 · 𝑖) → (-1↑𝑁) = 1)) |
23 | 22 | imp 406 | . 2 ⊢ ((𝑁 ∈ ℤ ∧ ∃𝑖 ∈ ℤ 𝑁 = (2 · 𝑖)) → (-1↑𝑁) = 1) |
24 | 4, 23 | sylbi 217 | 1 ⊢ (𝑁 ∈ Even → (-1↑𝑁) = 1) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∃wrex 3068 (class class class)co 7431 ℂcc 11151 0cc0 11153 1c1 11154 · cmul 11158 -cneg 11491 2c2 12319 ℤcz 12611 ↑cexp 14099 Even ceven 47549 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-uz 12877 df-seq 14040 df-exp 14100 df-even 47551 |
This theorem is referenced by: m1expoddALTV 47573 |
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